Two Discrete-Time Queues in Tandem

A discrete-time queueing problem involving two queues in tandem, with unit service time, is considered. Such a situation can arise within a packet-switching network. Arrivals at the queues occur from separate sources, which may be correlated. The joint arrivals from the two sources are assumed to be independently and identically distributed in each unit time interval. In addition, the output of the first queue enters the second queue. Both queues are assumed to have unlimited size, and the generating function of the steady state distribution of the two queue lengths is calculated, under the assumption that the mean combined input rate from the two sources is less than unity. The average waiting times are also derived, under the assumption that all arrivals take place at the end of a unit time interval. A particular example, in which the input to the first queue is geometrically distributed in each unit time interval, while the input from the source into the second queue is either 0 or 1, with fixed probabilities, is analyzed. The steady state probability that the length of the second queue exceeds
m
is calculated. The asymptotic behavior for
m \gg 1
changes significantly when a certain threshold, depending on the mean input rates, is reached. Numerical results are presented.